(CNN) -- The federal judge presiding over the Obama administration's lawsuit against Arizona has blocked a key provision of the state's new immigration law. But that is unlikely to change the public's mind.

After three months of bitter, unrelenting debate, polls show 60 percent of voters are strongly in favor of the law. Proponents of immigration reform, including the president, have largely ignored this support for the law, writing it off as anti-immigrant, if not downright bigoted.

But the truth is there can be no hope of passing an immigration overhaul opposed by 60 percent of the public. And perhaps instead of fighting the majority and questioning its motives, we ought to try listening to its concerns. Maybe, just maybe, that could point the way to the compromise we need on immigration.

Almost every major pollster has asked Americans about Arizona's SB 1070, which requires police to determine the immigration status of people they stop if they also suspect those people are in the U.S. illegally -- the provision blocked by the judge.

A Pew Research Center poll conducted in May found 59 percent of respondents in favor. A CBS News survey in early July found 57 percent calling it "about right." Other organizations have conducted surveys in Pennsylvania, where 52 percent approve; Georgia, where 68 percent want a similar law; Texas, where 69 percent want one, and of course Arizona itself, with 55 to 70 percent in favor of the law, depending on the poll.

Just as stark and equally consistent is the national polarization. SB 1070 and the Justice Department lawsuit against it have split Americans into two hostile camps.

Many more Republicans than Democrats support the measure: In many surveys, the difference is more than 30 percentage points.

More than two-thirds of the broader public oppose the boycotts aimed at blocking the law; between half and two-thirds oppose the federal lawsuit.

Eighty-six percent say immigration is going to play a somewhat or very important role in their voting decisions in November, and most of the 55 percent who say "very important" strongly support the law and oppose the suit.

But other numbers contradict this black and white picture. The reality is more complex.

A majority of Americans support SB 1070 and a majority supports comprehensive reform.--Tamar Jacoby

Whether talking to pollsters or online, people offer many different reasons for supporting SB 1070.

The most commonly heard argument, that "the law is the law," masks a range of attitudes toward illegal immigrants. Some proponents are convinced that by definition they are criminals.

"They've broken the law coming into the country, so they need to be arrested," a high school teacher in George told a pollster.

But others draw a distinction between criminals and those who have done no more than enter without papers.

"Look, I know that not everyone here illegally is up to no good," a Pennsylvania man who supports the measure wrote in a blog post.

According to a Rasmussen poll, at least half of those who endorse SB 1070 are at least "somewhat concerned" about potential civil rights violations.

Even more striking, a July Rasmussen poll that found 61 percent of respondents eager for their own legislature to enact an Arizona-like law also found 59 percent in favor of a welcoming federal immigration policy -- one that admits everyone except "national security threats, criminals and those who come to live off the U.S. welfare system."

Many who tell pollsters they endorse the Arizona act seem mainly concerned about sending a message to Washington. When Lake Research Partners asked people why they supported SB 1070, 52 percent chose "the state took action because the federal government had failed to solve the problem," compared with 28 percent who thought the law would reduce illegal immigration and 12 percent who thought it would reduce crime.

If anything, according to Rasmussen, Arizona's supporters are far angrier at the federal government (83 percent) than at illegal immigrants (12 percent).

This isn't to say there are no bigots among the law's supporters. There surely are, along with many who support the measure's stated aim of harassing illegal immigrants to the point that they leave the country voluntarily, a strategy the law's framers call "attrition through enforcement."

But these hardliners don't seem to be in the majority, because many of the law's proponents also support comprehensive immigration reform.

According to a poll commissioned by the Arizona Republic, 55 percent of repondents in the state endorse SB 1070, yet 62 percent support "allowing immigrants with no criminal record to remain in the United States." Indeed, according to Lake Research, those who support the Arizona law are actually more likely to support comprehensive reform, and more than three-quarters of them do.

This isn't as paradoxical as it seems.

After all, enforcement is an integral part of comprehensive reform -- one of three essential pillars, along with visas for future workers and legalization of the 11 million illegal immigrants already here.

True enough, as poll after poll has shown for many years, the pillar that matters most to the public is enforcement. When asked to choose, according to Rasmussen, 68 percent say gaining control of the border is most important; 24 percent choose legalizing illegal immigrants.

But that doesn't mean they think enforcement is enough.

From Rasmussen to Lake, all the research supports what I observed at a series of focus groups that my own organization commissioned this spring. Voters love enforcement and they will tolerate some temporary worker visas.

Put those two together, and Americans endorse the package enthusiastically. But when you ask, even among conservative Republicans, "Is anything missing?" several people in every group raise their hands.

"What are you going to do about the 11 million immigrants already in the country?" People just don't believe you can solve that piece of the puzzle with enforcement alone.

Where does this leave us? A majority of Americans support SB 1070 and a majority supports comprehensive reform. For many, it's not an either/or --it's a both/and.

If reformers played their cards right, they could enlist the 60 percent. But so far, they haven't played their cards right. On the contrary, they have only alienated the majority.

How to reverse this? President Clinton would have known instinctively.

The speech starts with "I feel your pain." Immigration reformers, starting with President Obama, need to reach out to Americans who support the Arizona law. They need to understand the public's anger. And they need to address its concerns.

This won't be an easy conversation. The administration's lawsuit makes it harder -- much harder. And it will take a lot of tact and humility on reformers' part.

But the polling doesn't lie. There's rich common ground there. And we won't solve the problem till we find it.

The opinions expressed in this commentary are solely those of Tamar Jacoby.